用Lax Friedrichs格式構造其Euler-Possion eq 的近似解。利用補償緊性框架得到\(\gamma=1\)時的收斂性和一致性。得到了\(L_\infty\)的全域性熵解。此處處理的是包含無界速度的初始條件，這與等熵情況不同。
對Possion equation 直接使用Green 函式法解出來。Euler-Possion 可以得到

\[\begin{align} \rho_t+m_x &=0 \\ m_t+(\frac{m^2}{\rho} +\rho )_x&=\rho(\int G_x(\rho-D) d\mu)-\frac{m}{\tau} \end{align} \]

初邊值條件檢視原文。
Theorem 1.2.

Suppose that the initial data \((\rho_0,m_0)\) and the given function \(D(x)\) and \(\tau\) satisfy the following conditions:

\[0\leq \rho_0 \leq C_0,|m_0| \leq \rho_0(C_0+|log \rho_0|),|D(x)| \leq C_0, 0<\tau_0 \leq \tau \]

Then the initial-boundary value problem for Euler-Possion has a global weak entropy solution \(\rho,m\)

and the following inequalities:

\[0<\rho<M,|m| \leq \rho(M+\log \rho) \]

as \(0<t<T\)

Lemma1 齊次方程Riemann 問題全域性解。
Lemma2 \(\Lambda={(\rho,m): w\leq w_0, z\geq z_0}\) is an invariant region. which means if. the Riemann data in \(\Lambda\) then the solution in \(\Lambda\) as well.

Entropy flux pair

The entropy-flux pair \((\eta,q)\) for \(\gamma=1\) are

 $ \eta=\rho^{1/(1-\xi^2)}e^{\xi/(1-xi^2)} m/ \rho ,  q=(m/\rho+\xi)\rho^{1/(1-\xi^2)}e^{\xi/(1-xi^2)} m/\rho $

Compact framework
if (1) \(0 \leq \rho^\epsilon \leq C ,|m^\epsilon| \leq (C+|ln \rho^\epsilon|)\)
(2) $$\eta_t(\rho^{\epsilon} ,m^\epsilon) +q_x(\rho^{\epsilon} ,m^\epsilon) $$ is compact in \(H^{-1}_{loc}\)
Then : exist subsequence \((\rho^{\epsilon},m^{\epsilon} ) \to (\rho,m)\) in \(L^p_{loc}\)

參考文獻

Li, Tian-Hong. “Convergence of the Lax–Friedrichs scheme for isothermal gas dynamics with semiconductor devices.” Zeitschrift für angewandte Mathematik und Physik ZAMP 57 (2005): 12-32.